Can the product of three complex numbers ever be real?

complex-numbers real-numbers examples-counterexamples

Say I have three numbers, $a,b,c\in\mathbb C$. I know that if $a$ were complex, for $abc$ to be real, $bc=\overline a$. Is it possible for $b,c$ to both be complex, or is it only possible for one to be, the other being a scalar?


Solution 1:

For example $z^3=1$, where $z\neq1.$

Id est, $$a=b=c=-\frac{1}{2}+\frac{\sqrt3}{2}i.$$

Solution 2:

I'm not sure I understood your question, but I suppose that the equality$$i\times(1+i)\times(1+i)=-2$$answers it.

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